The L2-approximation order of surface spline interpolation

We show that if the open, bounded domain Ω ⊂ Rd has a sufficiently smooth boundary and if the data function f is sufficiently smooth, then the Lp(Ω)-norm of the error between f and its surface spline interpolant is O(δγp+1/2) (1 ≤ p ≤ ∞), where γp := min{m,m − d/2 + d/p} and m is an integer parameter specifying the surface spline. In case p = 2, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the L2-approximation order of surface spline interpolation is m+ 1/2.